Local Linear Approximation for Functions of Several Variables

Functions of One Variable When we zoom in on a “sufficiently nice” function of one variable, we see a straight line.

Functions of two Variables

When we zoom in on a “sufficiently nice” function of two variables, we see a plane.

Describing the Tangent Plane What information do we need to describe this plane? The point (a,b) and the partials of f in the x- and y-directions. The equation of the plane is We can also write this in vector form. If we write and

For a general vector-valued function F:  n →  m and for a point p in  n, we want to find a linear function A p :  n →  m such that General Linear Approximations Why don’t we just subsume F(p) into A p ? Linear--- in the linear algebraic sense. In the expression we can think of the gradient as a scalar function on  2 : This function is linear.

Linear Functions A function A is said to be linear provided that Note that A (0) = 0, since A(x) = A (x+0) = A(x)+A(0). For a function A:  n →  m, these requirements are very prescriptive.

Linear Functions It can be shown that if A:  n →  m is linear, then In other words, the function A is just left-multiplication by a matrix. Thus we cheerfully confuse the function A, with the matrix that represents it!

Local Linear Approximation For all x, we have F(x)=A p (x-p)+F(p)+E(x) Where E(x) is the error committed by L p (x)= A p (x-p)+F(p) in approximating F(x)

Local Linear Approximation What can we say about the relationship between the matrix A p and the coordinate functions F 1, F 2, F 3,..., F m ? Quite a lot, actually... Fact: Suppose that F:  n →  m is given by coordinate functions F=(F 1, F 2,..., F m ) and all the partial derivatives of F exist at p   n, then... there is some matrix A p such that F can be approximated locally near p by the affine function

We Just Compute First, I ask you to believe that if L=(L 1,L 2,..., L n ) for all i and j with 1  i  n and 1  j  m This should not be too hard. Why? Think about tangent lines, think about tangent planes. Considering now the matrix formulation, what is the partial of L j with respect to x i ?

The Derivative of F at p (sometimes called the Jacobian Matrix of F at p)

Local Linear Approximation How should we think about the error function E(x)? It is easy to see that for all x, we can find E(x) so that F(x)=Df(p)(x-p)+F(p)+E(x). E(x) is the error committed by L p (x)= Df(p)(x-p)+F(p) in approximating F(x). So F will be “locally linear” if F(x)  Df(p)(x-p)+F(p) for all x “close” to p. How close does x have to be to p?

E(x) for One-Variable Functions But E(x)→0 is not enough, even for functions of one variable! What happens to E( x ) as x approaches p ? E(x) measures the vertical distance between f (x) and L p (x)

Differentiability of Vector Fields A vector-valued function F:  n →  m is said to be differentiable provided that there exists a function E :  n →  m such that for all x, F(x)=A p (x-p)+F(p)+E(x) where E(x) satisfies